On the harmonic oscillator on the Lobachevsky plane

Details On the harmonic oscillator on the Lobachevsky plane On the harmonic oscillator on the Lobachevsky plane

2007-09-24

arxiv.org/abs/0709.3697v1

Physics

Mathematical Physics

to appear in Russian Journal of Mathematical Physics (memorial volume in honor of Vladimir Geyler)

Scientific paper

10.1134/S1061920807040152

We introduce the harmonic oscillator on the Lobachevsky plane with the aid of the potential $V(r)=(a^2\omega^2/4)sinh(r/a)^2$ where $a$ is the curvature radius and $r$ is the geodesic distance from a fixed center. Thus the potential is rotationally symmetric and unbounded likewise as in the Euclidean case. The eigenvalue equation leads to the differential equation of spheroidal functions. We provide a basic numerical analysis of eigenvalues and eigenfunctions in the case when the value of the angular momentum, $m$, equals 0.

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